Singular values of compressions, restrictions and dilations

Abstract

For an operator A and a subspace E, denote by A:E the restriction of A to E and by AE the compression of A to E. A pair (S,T) of hermitian operators is said to be monotone if there exist two nondecreasing functions f, g and a hermitian Z such that S=f(Z) and T=g(Z). Using this notion: (1) We give a simple analytic characterisation of invertible operators X such that Sing(A:X(E))⩾Sing(A:E) for all subspaces E, where Sing(·) stands for the sequence of singular values. Equivalently, if A is hermitian we characterise invertible operators X satisfying Eig(AX(E))⩾Eig(AE) for all subspaces E, Eig(·) standing for the sequence of eigenvalues. (2) We prove many inequalities involving monotone pairs of positive operators. For instance we show that detAE·detBE⩽det(AB)E. In some other circumstances, the opposite inequality holds. We also show that pairs of hermitians can be dilated into monotone pairs.